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Kinetic Theory of Nucleation Kinetic Theory of Nucleation Eli Ruckenstein Gersh Berim Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20160601 International Standard Book Number-13: 978-1-4987-6284-7 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface xv Authors xix 1 Kinetic Theory of Liquid-to-Solid Nucleation 1 Introduction ...........................................................................................................................................................1 References ...............................................................................................................................................................2 1.1 A Possible Nucleation Type of Mechanism for the Growth of Small Aerosol Particles .................3 Ganesan Narsimhan and Eli Ruckenstein 1.1.1 Introduction ..............................................................................................................................3 1.1.2 Comparison between Rates of Formation and Dissociation of a Doublet Consisting of Equal-Size Aerosol Particles ..........................................................................4 1.1.2.1 Rate of Formation of a Doublet ..........................................................................4 1.1.2.2 Rate of Dissociation of a Doublet ......................................................................5 1.1.3 Two Alternate Mechanisms of Aerosol Growth ..................................................................9 1.1.4 Conclusions .............................................................................................................................10 References .............................................................................................................................................................11 1.2 A New Approach for the Prediction of the Rate of Nucleation in Liquids .....................................12 Ganesan Narsimhan and Eli Ruckenstein 1.2.1 Introduction ............................................................................................................................12 1.2.2 Structure of Molecular Clusters ...........................................................................................13 1.2.2.1 Amorphous Spherical Cluster ..........................................................................13 1.2.2.2 Spherical fcc Crystalline Cluster .....................................................................14 1.2.3 Interaction Potential of a Surface Monomer ......................................................................14 1.2.4 First Passage Time Analysis for Estimation of Rate of Dissociation of a Surface Monomer .................................................................................................................................15 1.2.5 Solubility ..................................................................................................................................19 1.2.6 Rate of Nucleation ..................................................................................................................20 1.2.7 Interfacial Tension of Macroscopic Clusters ......................................................................22 1.2.8 Comparison of the Prediction with the Classical Nucleation Theory ...........................24 1.2.9 Conclusions .............................................................................................................................28 References .............................................................................................................................................................28 1.3 A Kinetic Theory of Nucleation in Liquids ..........................................................................................30 Eli Ruckenstein and Bogdan Nowakowski 1.3.1 Introduction ............................................................................................................................30 1.3.2 Interaction between a Molecule and a Cluster ..................................................................30 1.3.3 Kinetics of Dissociation ........................................................................................................32 1.3.4 Rate of Condensation ............................................................................................................34 v 1.3.5 Critical Nucleus ......................................................................................................................35 1.3.6 Limit of Large Clusters and Thermodynamic Relations ..................................................35 1.3.7 Rate of Nucleation ..................................................................................................................36 1.3.8 Numerical Results and Comparison with the Classical Theory .....................................37 1.3.9 Conclusions .............................................................................................................................39 Appendix 1A: Asymptotic Expansion of the Equation for Radius of Critical Cluster (Equation 1.125)...40 References .............................................................................................................................................................41 1.4 Rate of Nucleation in Liquids for Face-Centered Cubic and Icosahedral Clusters .......................42 Bogdan Nowakowski and Eli Ruckenstein 1.4.1 Introduction ............................................................................................................................42 1.4.2 Theory of Nucleation—Mean Passage Time Approach ...................................................42 1.4.3 Structure of Clusters ..............................................................................................................44 1.4.4 Interaction of the Surface Atom with the Cluster .............................................................45 1.4.5 Results and Discussion ..........................................................................................................46 1.4.6 Conclusions .............................................................................................................................49 References .............................................................................................................................................................49 1.5 Comparison among Various Approaches to the Calculation of the Nucleation Rate ..................51 Bogdan Nowakowski and Eli Ruckenstein 1.5.1 Introduction ............................................................................................................................51 1.5.2 Zeldovich Expansion .............................................................................................................51 1.5.3 Goodrich and Modified Goodrich Expansions .................................................................52 1.5.4 Rate of Nucleation ..................................................................................................................53 1.5.5 Discussion ...............................................................................................................................54 References .............................................................................................................................................................54 1.6 A Kinetic Approach to the Theory of Heterogeneous Nucleation on Soluble Particles during the Deliquescence Stage .............................................................................................................55 Yuri Djikaev and Eli Ruckenstein 1.6.1 Introduction ............................................................................................................................55 1.6.2 Background of Deliquescence Theory ................................................................................56 1.6.2.1 Thermodynamics of Deliquescence ................................................................57 1.6.2.2 Kinetic Equation of Deliquescence .................................................................58 1.6.3 A New Approach to Kinetics of Deliquescence .................................................................59 1.6.3.1 Kinetics of Fluctuational Deliquescence ........................................................60 1.6.3.2 Kinetics of Barrierless Deliquescence .............................................................64 1.6.4 Numerical Evaluations ..........................................................................................................65 1.6.5 Conclusions .............................................................................................................................69 Acknowledgments ...............................................................................................................................................69 Appendix 1B: Derivation of Equation 1.185 ....................................................................................................69 Appendix 1C: Equilibrium Distribution of Composite Droplets ..................................................................70 Appendix 1D: Solution of Equation 1.185........................................................................................................72 Appendix 1E: Explicit Expressions for the Coefficients of Polynomial (Equation 1.208) .........................74 References .............................................................................................................................................................74 1.7 Effect of Solute–Solute and Solute–Solvent Interactions on Kinetics of Nucleation in Liquids ....76 Eli Ruckenstein and Gersh Berim 1.7.1 Introduction ............................................................................................................................76 1.7.2 Background .............................................................................................................................76 1.7.2.1 Interaction Potentials ........................................................................................76 1.7.2.2 Nucleation Rate ..................................................................................................78 1.7.2.3 Dissociation Rate................................................................................................79 1.7.2.4 Condensation Rate .............................................................................................81 1.7.3 Numerical Estimation of Effect of Solute–Solute and Solute–S olvent Interactions on Kinetics of Nucleation ............................................81 1.7.4 Conclusion ...............................................................................................................................84 References .............................................................................................................................................................84 vi Contents 1.8 Thermodynamics of Heterogeneous Crystal Nucleation in Contact and Immersion Modes .....85 Yuri Djikaev and Eli Ruckenstein 1.8.1 Introduction ............................................................................................................................85 1.8.2 Free Energy of Heterogeneous Formation of Crystal Nuclei in Contact and Immersion Modes ..........................................................................................................87 1.8.2.1 Foreign Particle Completely Immersed in the Liquid ..................................88 1.8.2.2 Foreign Particle in Contact with a Liquid–Vapor Interface ........................91 1.8.2.3 Comparison of Immersion and Contact Modes ...........................................94 1.8.3 Numerical Evaluations and Discussions ............................................................................95 1.8.4 Concluding Remarks .............................................................................................................97 Acknowledgments ...............................................................................................................................................98 Appendix 1F: Crystal Nuclei and Their Works of Formation ........................................................................98 References ...........................................................................................................................................................102 1.9 Correction to “Thermodynamics of Heterogeneous Crystal Nucleation in Contact and Immersion Modes” ........................................................................................................................103 Yuri Djikaev and Eli Ruckenstein 1.10 Homogeneous Crystal Nucleation in Droplets as a Method for Determining Line Tension of a Crystal–Liquid–Vapor Contact ....................................................................................................104 Yuri Djikaev and Eli Ruckenstein 1.10.1 Introduction ..........................................................................................................................104 1.10.2 Background: The Thermodynamics and Kinetics of Homogeneous Crystal Nucleation in a Droplet .......................................................................................................105 1.10.2.1 Free Energy of Homogeneous Formation of Crystal Nuclei in Surface-Stimulated and Volume-Based Modes ......................................105 1.10.2.2 Effect of Surface-Stimulated Mode on Kinetics of Crystal Nucleation ...109 1.10.3 Determination of Line Tension of an Ice–Water–Vapor Contact .................................110 1.10.3.1 Numerical Evaluations ....................................................................................111 1.10.4 Concluding Remarks ...........................................................................................................112 References ...........................................................................................................................................................112 2 Kinetic Theory of Vapor-to-Liquid Nucleation 115 Reference.............................................................................................................................................................115 2.1 A Kinetic Approach to the Theory of Nucleation in Gases .............................................................116 Bogdan Nowakowski and Eli Ruckenstein 2.1.1 Introduction ..........................................................................................................................116 2.1.2 Evaporation Rate from a Cluster ........................................................................................116 2.1.3 Rate of Growth of a Cluster ................................................................................................119 2.1.4 Rate of Nucleation ................................................................................................................119 2.1.5 Results and Comparison with the Classical Theory and Experiment .........................121 Appendix 2A ......................................................................................................................................................124 References ...........................................................................................................................................................125 2.2 A Unidimensional Fokker–Planck Approximation in the Treatment of Nucleation in Gases ............................................................................................................................................127 Eli Ruckenstein and Bogdan Nowakowski 2.2.1 Introduction ..........................................................................................................................127 2.2.2 Kinetic Equation in Energy Space .....................................................................................127 2.2.3 Rate of Evaporation of Molecules from a Cluster............................................................131 2.2.4 Rate of Growth of a Cluster ................................................................................................131 2.2.5 Rate of Nucleation ................................................................................................................132 2.2.6 Results and Discussion ........................................................................................................133 2.2.7 Conclusion .............................................................................................................................135 References ...........................................................................................................................................................135 Contents vii 2.3 Homogeneous Nucleation in Gases: A Three-Dimensional Fokker–Planck Equation for Evaporation from Clusters .............................................................................................................136 Bogdan Nowakowski and Eli Ruckenstein 2.3.1 Introduction ..........................................................................................................................136 2.3.2 Equation for Probability Density Distribution Function of Energy ............................136 2.3.3 Kinetics of Growth (Evaporation) of a Single Cluster ....................................................140 2.3.4 Rate of Nucleation ................................................................................................................142 2.3.5 Numerical Results and Discussion ....................................................................................143 References ...........................................................................................................................................................145 2.4 Time-Dependent Cluster Distribution in Nucleation ......................................................................146 Bogdan Nowakowski and Eli Ruckenstein 2.4.1 Introduction ..........................................................................................................................146 2.4.2 Equations for the Population of Clusters ..........................................................................146 2.4.3 Solution of Time-Dependent Equations ...........................................................................149 2.4.4 Results of Numerical Calculations ....................................................................................150 2.4.5 Conclusion .............................................................................................................................154 References ...........................................................................................................................................................155 2.5 Kinetic Theory of Nucleation Based on a First Passage Time Analysis: Improvement by the Density-Functional Theory ......................................................................................................156 Yuri Djikaev and Eli Ruckenstein 2.5.1 Introduction ..........................................................................................................................156 2.5.2 Theoretical Basis ...................................................................................................................156 2.5.2.1 RNN Approach to Kinetics of Nucleation ...................................................156 2.5.2.2 DFT Methods in Nucleation Theory .............................................................159 2.5.2.3 Improvement of RNN Kinetic Theory of Nucleation by DFT Methods ....160 2.5.3 Numerical Results ................................................................................................................161 2.5.4 Concluding Remarks ...........................................................................................................163 Acknowledgments .............................................................................................................................................163 References ...........................................................................................................................................................163 2.6 Kinetic Theory of Binary Nucleation Based on a First Passage Time Analysis ...........................165 Yuri Djikaev and Eli Ruckenstein 2.6.1 Introduction ..........................................................................................................................165 2.6.2 Kinetics of Binary Nucleation Based on First Passage Time Analysis .........................166 2.6.2.1 Thermodynamics of Binary Nucleation Based on Capillarity Approximation .................................................................................................167 2.6.2.2 A New Approach to Kinetics of Binary Nucleation ....................................168 2.6.3 Model Binary System with a Surface Active Component ..............................................174 2.6.3.1 Density Functional Formalism ......................................................................175 2.6.4 Numerical Evaluations ........................................................................................................177 2.6.5 Concluding Remarks ...........................................................................................................179 Acknowledgments .............................................................................................................................................180 Appendix 2B: Derivation of Equation 2.183 ..................................................................................................180 Appendix 2C: Equilibrium Distribution of Binary Clusters ........................................................................181 Appendix 2D: Steady-State Solution of Equation 2.183 ...............................................................................182 References ...........................................................................................................................................................184 2.7 A Kinetic Treatment of Heterogeneous Nucleation .........................................................................186 Bogdan Nowakowski and Eli Ruckenstein 2.7.1 Introduction ..........................................................................................................................186 2.7.2 Rate of Evaporation from the Cluster ...............................................................................186 2.7.3 Growth of Cluster.................................................................................................................188 2.7.4 Rate of Nucleation ................................................................................................................189 2.7.5 Results of Calculations and Discussion ............................................................................191 viii Contents Appendix 2E: Interaction Potential between a Molecule and a Spherical Cap Cluster ............................192 References ...........................................................................................................................................................193 2.8 New Approach to the Kinetics of Heterogeneous Unary Nucleation on Liquid Aerosols of a Binary Solution ...............................................................................................................................194 Yuri Djikaev and Eli Ruckenstein 2.8.1 Introduction ..........................................................................................................................194 2.8.2 Kinetics of Nucleation .........................................................................................................196 2.8.3 A New Approach to Kinetics of Heterogeneous Nucleation .........................................198 2.8.4 Numerical Evaluations ........................................................................................................200 2.8.5 Conclusions ...........................................................................................................................203 Acknowledgments .............................................................................................................................................203 Appendix 2F: Kinetic Equation of Heterogeneous Nucleation ...................................................................203 Appendix 2G: Emission Rate via a First Passage Time Analysis .................................................................204 Appendix 2H: Equilibrium Distribution of Heterogeneously Formed Droplets ......................................206 References ...........................................................................................................................................................207 2.9 Kinetics of Heterogeneous Nucleation on a Rough Surface: Nucleation of a Liquid Phase in Nanocavities .......................................................................................................................................208 Eli Ruckenstein and Gersh Berim 2.9.1 Introduction ..........................................................................................................................208 2.9.2 The System and the Interaction Potentials .......................................................................209 2.9.3 Basic Equations of the Kinetic Theory ..............................................................................210 2.9.4 Results ....................................................................................................................................213 2.9.5 Discussion .............................................................................................................................216 Acknowledgments .............................................................................................................................................217 References ...........................................................................................................................................................217 2.10 Kinetic Theory of Heterogeneous Nucleation: Effect of Nonuniform Density in the Nuclei ...............................................................................................................................218 Gersh Berim and Eli Ruckenstein 2.10.1 Introduction ..........................................................................................................................218 2.10.2 The System, the Interaction Potentials, and the Definition of the Nucleus ................218 2.10.3 Basic Equations of KNT ......................................................................................................220 2.10.4 Results and Conclusion .......................................................................................................221 Appendix 2I: Contributions to the Free Energy of the System and Derivation of Euler–Lagrange Equation for FDD ...................................................................................................................................224 Appendix 2J: Basic Equations of KNT ............................................................................................................225 References ...........................................................................................................................................................228 3 Application of the Kinetic Nucleation Theory to Protein Folding 229 References ...........................................................................................................................................................229 3.1 Model for the Nucleation Mechanism of Protein Folding ..............................................................231 Yuri Djikaev and Eli Ruckenstein 3.1.1 Introduction ..........................................................................................................................231 3.1.2 Heteropolymer Chain as a Protein Model and a CNT-Based Model for the Nucleation Mechanism of Protein Folding .........................................................232 3.1.2.1 Heteropolymer as a Protein Model ...............................................................232 3.1.3 A Kinetic Approach to the Nucleation Mechanism of Protein Folding ......................234 3.1.3.1 Potential Well around a Cluster of Native Residues ...................................234 3.1.3.2 Determination of Emission and Absorption Rates .....................................238 3.1.3.3 Equilibrium Distribution and Steady-State Nucleation Rate ....................238 3.1.3.4 Evaluation of Protein Folding Time ..............................................................241 3.1.4 Numerical Evaluations ........................................................................................................241 3.1.5 Conclusions ...........................................................................................................................243 Appendix 3A: Derivation of Equation 3.11 from Equation 3.9 ...................................................................245 Contents ix

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